Hi Engineer4Christ,
I can imagine this all seems pretty overwhelming. I find it helpful to separate out one circumstance from another and then put it all back together.
As far as your current girlfriend goes, you will need to have an honest conversation with her. The best thing you can do as a man is take full responsibility for what you have done. I am not sure if you are wanting to try to continue on in this relationship or not, but I anticipate that it is going to be difficult because she will have a difficult time moving past the betrayal and that this child could be an ongoing reminder to her of your infidelity.
I think you need to assess what you want here as opposed to what you think you should do. Feeling like you should stay in the relationship is not the right reason to stay. You should stay because you love her and intend to continue pursuing relationship with her. This is going to mean a tremendous amount of work on your part, though, and even apart from that she is going to have to decide if she can continue on.
As far as the child goes, you will have to communicate with the mother to determine what is best for him/her. Maybe it means that you are the primary caretaker, maybe it means that she is. Either way I would plan on being a part of financially supporting the child.
I would also recommend getting into some counseling, especially to work through the things that led you into this affair. Oftentimes there is something going on at the heart level that really needs to be addressed in order to be able to have healthy relationships going forward.